Taking the partial trace gives us a sum of the form below with at least two nonzero terms.
$$\rho_A = \sum_k \lambda_k \vert k\rangle\langle k\vert$$

This is a diagonal matrix with rank 2 or more and is hence a mixed state.

Is there a similar argument one can make for the case where $\rho$ is a mixed entangled state? Alternatively, if this is not true, can one provide a counter example for which the reduced state is pure but the state $\rho$ is still entangled?

$\begingroup$What do you mean by "similar argument"?$\endgroup$
– Norbert SchuchMar 27 at 8:42

$\begingroup$@Norbert Schuch I mean an argument that proves that the reduced state of a mixed entangled state is always mixed.$\endgroup$
– user1936752Mar 27 at 9:30

$\begingroup$But this doesn't teach us anything: The reduced state of a non-entangled mixed states will also be mixed (with few exceptions).$\endgroup$
– Norbert SchuchMar 27 at 9:44

$\begingroup$@NorbertSchuch yes, that's true. But I was wondering if either one can prove the statement that all entangled mixed states have reduced states that are mixed or find a counterexample of an entangled mixed state with a reduced state that is pure.$\endgroup$
– user1936752Mar 27 at 10:50

$\begingroup$Then, PLEASE, ask that CLEARLY in your question! -- Other than that, the statement is true.$\endgroup$
– Norbert SchuchMar 27 at 12:57